Write ev*:LX→Xev_* : L X \to X for the evaluation map at the basepoint of the loops.

For [α]∈Hi(LX)[\alpha] \in H_i(L X) and [β]∈Hj(LX)[\beta] \in H_j(L X) we can find representatives α\alpha and β\beta such that ev(α)ev(\alpha) and ev(β)ev(\beta) intersect transversally. There is then an ((i+j)−dimX)((i+j)-dim X)-chain α⋅β\alpha \cdot \beta such that ev(α⋅β)ev(\alpha \cdot \beta) is the chain given by that intersection: above x∈ev(α⋅β)x \in ev(\alpha \cdot \beta) this is the loop obtained by concatenating αx\alpha_x and βx\beta_x at their common basepoint. The string product is then defined using such representatives by

For ℬ={A,B,⋯}\mathcal{B} = \{A, B , \cdots\} a collection of oriented compact submanifolds write PX(A,B)P_X(A,B) for the path space of paths in XX that start in A⊂XA \subset X and end in B⊂XB \subset X.

Theorem

The tuple (H•(LM,ℚ),{H•(PX(A,B),ℚ)}A,B∈ℬ)(H_\bullet(L M, \mathbb{Q}), \{H_\bullet(P_X(A,B), \mathbb{Q})\}_{A,B \in \mathcal{B}}) carries the structure of a dd-dimensional HCFT with positive boundary and set of branesℬ\mathcal{B}, such that the correlators in the closed sector are the standard string topology operation.

For the string product and the BV-operator this extension has been known early on, it yields a homotopy BV algebra considered around page 101 of

Scott Wilson, On the Algebra and Geometry of a Manifold’s Chains and Cochains (2005) (pdf)

Evidence for the existence of the TCFT version by exhibiting a dg-category that looks like it ought to be the dg-category of string-topology branes (hence ought to correspond to the TCFT under the suitable version of the TCFT-version of the cobordism hypothesis) is discussed in